The question is as follows:
By defining a suitable Euclidean ring, prove that there exist two integers a and b such that 512a + 972b = 4, explaining your reasoning clearly. (Note: you do not need to obtain specific values of a and b.)
I assume that I need to somehow use the fact that if $d=\gcd(a, b)$ then $\exists\lambda, \mu$ st $d=\lambda a+ \mu b$
So if I manage to show that $4=\gcd(512, 972)$, then I can use the above statement.
How do I show that $4=\gcd(512, 972)$ and how can I define a suitable Euclidean Ring?
Obviously $4|512$ and $4|972$ and I thought maybe I could consider $4\mathbb{Z}$ to be the Euclidean Ring...but I don't know how to proceed from here.
Use the Euclidean algorithm for $512/4=128$ and $972/4=243$ in the Euclidean ring $\mathbb{Z}$. We have $gcd(128,243)=1$, and the Euclidean algorithm gives $$128\cdot(-112)+243\cdot 59=1.$$ Then multiply this equation by $4$.